Using the South Pole Telescope (SPT), we have discovered the most massive
known galaxy cluster at z > 1, SPT-CL J2106-5844. In addition to producing a
strong Sunyaev-Zel'dovich effect signal, this system is a luminous X-ray source
and its numerous constituent galaxies display spatial and color clustering, all
indicating the presence of a massive galaxy cluster. VLT and Magellan
spectroscopy of 18 member galaxies shows that the cluster is at z =
1.132^+0.002_-0.003. Chandra observations obtained through a combined HRC-ACIS
GTO program reveal an X-ray spectrum with an Fe K line redshifted by z = 1.18
+/- 0.03. These redshifts are consistent with galaxy colors in extensive
optical, near-infrared, and mid-infrared imaging. SPT-CL J2106-5844 displays
extreme X-ray properties for a cluster, having a core-excluded temperature of
kT = 11.0^+2.6_-1.9 keV and a luminosity (within r_500) of L_X (0.5 - 2.0 keV)
= (13.9 +/- 1.0) x 10^44 erg/s. The combined mass estimate from measurements of
the Sunyaev-Zel'dovich effect and X-ray data is M_200 = (1.27 +/- 0.21) x 10^15
M_sun. The discovery of such a massive gravitationally collapsed system at high
redshift provides an interesting laboratory for galaxy formation and evolution,
and is a powerful probe of extreme perturbations of the primordial matter
density field. We discuss the latter, determining that, under the assumption of
LambdaCDM cosmology with only Gaussian perturbations, there is only a 7% chance
of finding a galaxy cluster similar to SPT-CL J2106-5844 in the 2500 deg^2 SPT
survey region, and that only one such galaxy cluster is expected in the entire
sky.

This paper looks at the detection of a large (>1015) cluster at z~1.1 by the South Pole Telescope, and its cosmological implications. Sections 2 and 3 give a nice description of the detection and observational methods, but I’m a little concerned about section 4. One of the key claims of this work is that “there is a 7% chance of finding a cluster at least as massive and at a redshift at least as high”.

Quantifying the extreme nature of a single variable, such as mass, would be a meaningful frequentist statement (although for those evangelical Bayesians out there, no such statement exists, but bear with me…). I’m just a bit concerned by this double condition which is enforced – using both mass and redshift.

For example, take a sample population of people in the UK, only one person can claim to be the shortest, and one the heaviest. But many can claim that ‘no one is both shorter and heavier than me’. Perhaps each city has one such person. We don’t know how many of these people there will be, or their statistical significance, until we better understand the relationship linking the two variables. What is really needed is a single measure, such as BMI in this analogy.

An appropriate statistic needs to invoke the mass-z relationship, (as depicted in Fig 5 of 1101.1290), which then allows us to compute whether one cluster is more extreme than another. This analysis will likely generate a result significantly greater than the quoted 7%, as it opens up a much larger area of parameter space.

I guess I don't see the fundamental problem Fergus mentioned. After all, the upper bound predicted by a cosmological model (say best-fit LCDM) would, in this case, correspond to some curve in the M-z plane (for example, 95% curve or, to be more precise, curve bounding where 95% of LCDM models allowed by cosmological data contain 95% of their clusters). Whatever this curve shows, cluster or clusters above it can, given sufficient statistical significance, rule out LCDM. As mentioned in the previous post, this can happen in multiple ways, but that's fine.

This to me seems similar as comparing constraints from some new dataset to an old dataset in N-dimensional parameter space. The new set can be inconsistent with the old one in many different ways, with 'many' corresponding to different directions of non-overlap of constraints in the parameter space. Moreover, multiple new datasets may all be inconsistent with the old data, in principle.

At any rate, these are interesting new papers by SPT. Too bad LCDM survives again.

Imagine everyone in the world has the same BMI. Now select one person at random.

No one is "shorter and heavier" than this person, in the whole world. So we have a remarkable 0% expectation of finding someone shorter and heavier when expanding our survey (or conducting our MCMC). Is this statistically significant? Have we found a special person? Certainly not, this same result arises for any individual, and it merely reflects the strong correlation which exists between height and mass.

But anyway, we're agreed that LCDM remains intact. And yes I certainly don't mean to detract from the importance of the SPT results.

What would you say is the best thing to use as a cluster mass index? Weight over height2 obviously won't work. Number per dz and d(lnM) sort of does what you are talking about, but should this include the SPT selection function? If not, then you have to start worrying about stuff at reionization as possibly being there. A big advantage of >M,>z is it is easy to calculate, easy to explain.

From the results of the other SPT paper, you can guess that this object is not shockingly rare in a raw abundance sense: there are lower z objects that lightly stress LCDM by a comparable amount (setting aside that the X-ray derived mass for this big one is a bit higher). If LCDM as a whole doesn't seem broken and there are 1 or 2 objects that are about as rare, then you know that this object can't be a huge outlier. However, even after saying all this, I am still amazed that something this big is sitting out at z>1 (I know, it borders on irrational).

To continue with Fergus' example, imagine a town with people with the same (or similar) BMI, where therefore 'every person is special' in having a more extreme (mass, 1/height2) pair than anyone else.

Then the thing to do is compare these properties with theoretical prediction for the population of that size. That's the only comparison that matters. So for example, if by chance the theoretical model always predicts BMIconst with small uncertainty around this constant value, then there are two possible outcomes: either almost every person is an outlier, or almost nobody is. Then again, if the predicted BMI is variable (i.e. if the model predicts a relation in the (mass, height) with a slope different from 1/2 which is the BMI=const value), then only some (or a few) people may be unusual by the (mass, 1/height2) criterion.

So even if we don't like the statistic for some subjective reason, it's the Monte-Carlo comparison to theory that makes it fair. That's at least how I understand these approaches.

What would you say is the best thing to use as a cluster mass index? Weight over height2 obviously won't work. Number per dz and d(lnM) sort of does what you are talking about, but should this include the SPT selection function? If not, then you have to start worrying about stuff at reionization as possibly being there.
.

Yes it would be quite neat to define a 'Cluster Mass Index', m/f(z), but as you say, f(z) is not quite as simple as zn! But since we have a natural reference "height" in this case, z=0, perhaps it would be informative to rescale masses to what we expect them to be at z=0. Looking at it this way, I wonder if SPT-CL J2106−5844 is more massive (today) than any other known object?!

Gil Holder wrote:

A big advantage of >M,>z is it is easy to calculate, easy to explain.
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Ah, sorry I didn't mean to imply we should never perform this type of two-parameter cut, like p(m>M,z>Z). It only becomes pathological when, as in this example, the constants M and Z are defined after looking at the data. At least one of the two (ideally both) ought to be predefined. Otherwise we can get these rather nonsensical "every person is an outlier" scenarios! Or start worrying that last week's lottery numbers were a freak occurrence.